Theorem sbceqbidf 29321

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)

Hypotheses

Ref	Expression
sbceqbidf.1	|- F/ x ph
sbceqbidf.2	|- ( ph -> A = B )
sbceqbidf.3	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbceqbidf	|- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )

Proof of Theorem sbceqbidf

Step	Hyp	Ref	Expression
1		sbceqbidf.2	. . 3 |- ( ph -> A = B )
2		sbceqbidf.1	. . . 4 |- F/ x ph
3		sbceqbidf.3	. . . 4 |- ( ph -> ( ps <-> ch ) )
4	2, 3	abbid 2740	. . 3 |- ( ph -> { x | ps } = { x | ch } )
5	1, 4	eleq12d 2695	. 2 |- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
6		df-sbc 3436	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
7		df-sbc 3436	. 2 |- ( [. B / x ]. ch <-> B e. { x | ch } )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)